[N,k,chi] = [75,5,Mod(7,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.7");
S:= CuspForms(chi, 5);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
\(n\)
\(26\)
\(52\)
\(\chi(n)\)
\(1\)
\(\beta_{2}\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 60T_{2}^{5} + 1973T_{2}^{4} + 3300T_{2}^{3} + 1800T_{2}^{2} - 31560T_{2} + 276676 \)
T2^8 + 60*T2^5 + 1973*T2^4 + 3300*T2^3 + 1800*T2^2 - 31560*T2 + 276676
acting on \(S_{5}^{\mathrm{new}}(75, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{8} + 60 T^{5} + 1973 T^{4} + \cdots + 276676 \)
T^8 + 60*T^5 + 1973*T^4 + 3300*T^3 + 1800*T^2 - 31560*T + 276676
$3$
\( (T^{4} + 729)^{2} \)
(T^4 + 729)^2
$5$
\( T^{8} \)
T^8
$7$
\( T^{8} + 20 T^{7} + \cdots + 30620728960000 \)
T^8 + 20*T^7 + 200*T^6 - 247720*T^5 + 37960804*T^4 - 864647360*T^3 + 5797491200*T^2 + 595858048000*T + 30620728960000
$11$
\( (T^{4} + 144 T^{3} - 16742 T^{2} + \cdots + 27154144)^{2} \)
(T^4 + 144*T^3 - 16742*T^2 - 174672*T + 27154144)^2
$13$
\( T^{8} - 340 T^{7} + \cdots + 158448579136 \)
T^8 - 340*T^7 + 57800*T^6 - 866920*T^5 + 1026388*T^4 + 686031040*T^3 + 83199363200*T^2 + 162375003520*T + 158448579136
$17$
\( T^{8} + \cdots + 125636659418176 \)
T^8 + 900*T^7 + 405000*T^6 + 87175800*T^5 + 9840452948*T^4 + 208418436000*T^3 + 2003201280000*T^2 - 22435486041600*T + 125636659418176
$19$
\( T^{8} + 567288 T^{6} + \cdots + 82\!\cdots\!00 \)
T^8 + 567288*T^6 + 86280475536*T^4 + 4785962831539200*T^2 + 82644855537346560000
$23$
\( T^{8} - 1560 T^{7} + \cdots + 40\!\cdots\!00 \)
T^8 - 1560*T^7 + 1216800*T^6 - 360543840*T^5 + 130274568464*T^4 - 123487954149120*T^3 + 99119043846604800*T^2 - 28345484241040896000*T + 4053037870817896960000
$29$
\( T^{8} + 1665012 T^{6} + \cdots + 23\!\cdots\!00 \)
T^8 + 1665012*T^6 + 990525952836*T^4 + 252617889539462400*T^2 + 23493392388495936000000
$31$
\( (T^{4} + 256 T^{3} + \cdots + 388673200000)^{2} \)
(T^4 + 256*T^3 - 2345016*T^2 - 890254400*T + 388673200000)^2
$37$
\( T^{8} - 3820 T^{7} + \cdots + 60\!\cdots\!00 \)
T^8 - 3820*T^7 + 7296200*T^6 - 3815110360*T^5 + 5587793280724*T^4 - 15126290999355200*T^3 + 24290307812208080000*T^2 - 17096356988493531440000*T + 6016503054175207081960000
$41$
\( (T^{4} + 1356 T^{3} + \cdots - 1195607024000)^{2} \)
(T^4 + 1356*T^3 - 2977076*T^2 - 4901830080*T - 1195607024000)^2
$43$
\( T^{8} - 1240 T^{7} + \cdots + 37\!\cdots\!76 \)
T^8 - 1240*T^7 + 768800*T^6 + 83727680*T^5 + 6510806848*T^4 - 1281494389760*T^3 + 88706937651200*T^2 - 2567722303815680*T + 37162808254283776
$47$
\( T^{8} + 4800 T^{7} + \cdots + 32\!\cdots\!76 \)
T^8 + 4800*T^7 + 11520000*T^6 + 10603958400*T^5 + 5100280623248*T^4 + 642578082163200*T^3 + 551108889031680000*T^2 + 597607201890508185600*T + 324014341683820183883776
$53$
\( T^{8} + 1020 T^{7} + \cdots + 29\!\cdots\!00 \)
T^8 + 1020*T^7 + 520200*T^6 - 4336611000*T^5 + 81018899379524*T^4 + 46636261234848480*T^3 + 14826052484977564800*T^2 - 29820754031681595552000*T + 29990362300386791566240000
$59$
\( T^{8} + 28672428 T^{6} + \cdots + 12\!\cdots\!00 \)
T^8 + 28672428*T^6 + 282101285468196*T^4 + 1085431868712613584000*T^2 + 1218010944198742985139840000
$61$
\( (T^{4} + 2380 T^{3} + \cdots - 51045861284864)^{2} \)
(T^4 + 2380*T^3 - 19737588*T^2 - 70624872320*T - 51045861284864)^2
$67$
\( T^{8} - 8920 T^{7} + \cdots + 37\!\cdots\!96 \)
T^8 - 8920*T^7 + 39783200*T^6 - 60681760960*T^5 + 844296225464128*T^4 - 7107101264762209280*T^3 + 31647675741397699788800*T^2 - 48588724545772457065840640*T + 37299171229449104193771421696
$71$
\( (T^{4} - 3768 T^{3} + \cdots - 4392786466304)^{2} \)
(T^4 - 3768*T^3 - 41656016*T^2 + 67387285248*T - 4392786466304)^2
$73$
\( T^{8} + 11600 T^{7} + \cdots + 48\!\cdots\!00 \)
T^8 + 11600*T^7 + 67280000*T^6 + 131415261920*T^5 + 149959499848264*T^4 + 501709456452063040*T^3 + 4365540077803830387200*T^2 + 6505419271620224632240000*T + 4847107018297945884210250000
$79$
\( T^{8} + 118621200 T^{6} + \cdots + 60\!\cdots\!00 \)
T^8 + 118621200*T^6 + 5076160449480000*T^4 + 92334854675016864000000*T^2 + 600919454860090745760000000000
$83$
\( T^{8} - 32400 T^{7} + \cdots + 23\!\cdots\!56 \)
T^8 - 32400*T^7 + 524880000*T^6 - 4476479960640*T^5 + 22445139154298768*T^4 - 56972560249412889600*T^3 + 84342731778388249804800*T^2 - 63117457768620682168565760*T + 23616815528581306505462419456
$89$
\( T^{8} + 128964168 T^{6} + \cdots + 22\!\cdots\!00 \)
T^8 + 128964168*T^6 + 5516575607843856*T^4 + 82917133460560785945600*T^2 + 225437496827007286630656000000
$97$
\( T^{8} + 58640 T^{7} + \cdots + 23\!\cdots\!96 \)
T^8 + 58640*T^7 + 1719324800*T^6 + 29283250529120*T^5 + 314465299384051528*T^4 + 2052006934191997840960*T^3 + 8416079426197616809587200*T^2 + 19856885564573538680863799680*T + 23425153468561774348926724404496
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